Constants library

292 values, each with its units, its uncertainty, and where it came from.

Universal & Atomic (47)Electromagnetic (31)Thermodynamic (37)Astronomical (47)Material Properties (69)Chemistry (45)Mathematical (16)

Universal & Atomic 47

Speed of light in vacuum exact

c = 299,792,458\ \text{m/s}

m/sThe invariant speed of light in vacuum, exactly 299 792 458 m/s since 1983 — the definition that now fixes the length of the metre.

Speed of light squared (mass-energy conversion factor) exact

c^{2} = 8.987551787368176 \times 10^{16}\ \text{J/kg}

J/kgThe exchange rate between mass and energy in E = mc²: 8.988 × 10¹⁶ joules locked in every kilogram of rest mass.

Planck constant exact

h = 6.62607015 \times 10^{-34}\ \text{J}{\cdot}\text{s}

J·sThe quantum of action, exactly 6.626 070 15 × 10⁻³⁴ J·s — the constant that has defined the kilogram since 2019.

Reduced Planck constant (Dirac constant) exact

\hbar = 1.054571817 \times 10^{-34}\ \text{J}{\cdot}\text{s}

J·sPlanck's constant divided by 2π, 1.054 571 817 × 10⁻³⁴ J·s — the natural quantum of angular momentum and spin.

Newtonian constant of gravitation measured

G = 6.6743 \times 10^{-11}\ \text{m}^{3}\text{/(kg}{\cdot}\text{s}^{2}\text{)}

m³/(kg·s²)The coupling strength of gravity, 6.674 30 × 10⁻¹¹ m³/(kg·s²) — the worst-measured constant in all of physics.

Elementary charge exact

e = 1.602176634 \times 10^{-19}\ \text{C}

CThe charge of a proton, exactly 1.602 176 634 × 10⁻¹⁹ C — the quantum of free charge and the SI definition of the ampere.

Electronvolt (in joules) exact

\mathrm{eV} = 1.602176634 \times 10^{-19}\ \text{J}

JThe energy one electron gains crossing one volt: exactly 1.602 176 634 × 10⁻¹⁹ J, the working currency of atomic physics.

Standard acceleration of gravity exact

g_{n} = 9.80665\ \text{m/s}^{2}

m/s²The conventional value of free-fall acceleration, exactly 9.806 65 m/s² — a defined reference, not a measurement of your local g.

Avogadro constant exact

N_{\mathrm{A}} = 6.02214076 \times 10^{23}\ \text{mol}^{-1}

mol⁻¹Exactly 6.022 140 76 × 10²³ entities per mole — the fixed number that has defined the mole since the 2019 SI revision.

Atomic mass constant (unified atomic mass unit) measured

m_{\mathrm{u}} = 1.66053906892 \times 10^{-27}\ \text{kg}

kgOne twelfth of the mass of a free carbon-12 atom at rest, 1.660 539 069 × 10⁻²⁷ kg — the dalton used in every mass spectrum.

Atomic mass constant energy equivalent measured

m_{\mathrm{u}} c^{2} = 1.49241808768 \times 10^{-10}\ \text{J}

JThe rest energy of one dalton, 1.492 418 088 × 10⁻¹⁰ J or 931.494 MeV — the conversion factor behind every mass-defect calculation.

Electron mass measured

m_{\mathrm{e}} = 9.1093837139 \times 10^{-31}\ \text{kg}

kgThe rest mass of the electron, 9.109 383 714 × 10⁻³¹ kg — equivalently 5.485 799 091 × 10⁻⁴ u or 0.510 999 MeV/c².

Electron mass energy equivalent measured

m_{\mathrm{e}} c^{2} = 8.187105788 \times 10^{-14}\ \text{J}

JThe electron's rest energy, 8.187 105 788 × 10⁻¹⁴ J or 510.999 keV — the photon energy of every positron annihilation.

Proton mass measured

m_{\mathrm{p}} = 1.67262192595 \times 10^{-27}\ \text{kg}

kgThe rest mass of the proton, 1.672 621 926 × 10⁻²⁷ kg — 1.007 276 u, 938.272 MeV/c², and 1836 times the electron.

Proton mass energy equivalent measured

m_{\mathrm{p}} c^{2} = 1.50327761802 \times 10^{-10}\ \text{J}

JThe proton's rest energy, 1.503 277 618 × 10⁻¹⁰ J or 938.272 MeV — the yardstick for accelerator and nuclear energies.

Neutron mass measured

m_{\mathrm{n}} = 1.67492750056 \times 10^{-27}\ \text{kg}

kgThe rest mass of the neutron, 1.674 927 501 × 10⁻²⁷ kg — 1.008 665 u or 939.565 MeV/c², just heavier than the proton.

Neutron mass energy equivalent measured

m_{\mathrm{n}} c^{2} = 1.50534976514 \times 10^{-10}\ \text{J}

JThe neutron's rest energy, 1.505 349 765 × 10⁻¹⁰ J or 939.565 MeV — exceeding the proton's by the 1.293 MeV that drives beta decay.

Muon mass measured

m_{\mu} = 1.883531627 \times 10^{-28}\ \text{kg}

kgThe rest mass of the muon, 1.883 531 627 × 10⁻²⁸ kg or 105.658 MeV/c² — 207 electrons in one unstable package.

Tau lepton mass measured

m_{\tau} = 3.16754 \times 10^{-27}\ \text{kg}

kgThe rest mass of the tau lepton, 3.167 54 × 10⁻²⁷ kg or 1776.86 MeV/c² — heavier than a proton, and the shortest-lived lepton.

Deuteron mass measured

m_{\mathrm{d}} = 3.3435837768 \times 10^{-27}\ \text{kg}

kgThe mass of the deuteron, one proton bound to one neutron: 3.343 583 777 × 10⁻²⁷ kg, or 2.013 553 u and 1875.613 MeV/c².

Alpha particle mass measured

m_{\alpha} = 6.644657345 \times 10^{-27}\ \text{kg}

kgThe mass of the helium-4 nucleus, 6.644 657 345 × 10⁻²⁷ kg — 4.001 506 u, 3727.379 MeV/c², and the most tightly bound light nucleus.

Proton-electron mass ratio measured

m_{\mathrm{p}}/m_{\mathrm{e}} = 1836.152673426

dimensionlessThe proton outweighs the electron by 1836.152 673 4 — a pure number, known to 17 parts per trillion, that shapes all of chemistry.

Neutron-proton mass ratio measured

m_{\mathrm{n}}/m_{\mathrm{p}} = 1.00137841946

dimensionlessThe neutron is heavier than the proton by just 0.1378 % — the 1.293 MeV difference that makes free neutrons decay and stars burn.

Muon-electron mass ratio measured

m_{\mu}/m_{\mathrm{e}} = 206.7682827

dimensionlessThe muon is 206.768 times heavier than the electron — the same particle in every respect except mass, and nobody knows why.

Fine-structure constant measured

\alpha = 0.0072973525643

dimensionlessThe dimensionless strength of the electromagnetic interaction, 0.007 297 352 564 — roughly 1/137, and pure number with no units at all.

Inverse fine-structure constant measured

\alpha^{-1} = 137.035999177

dimensionlessThe reciprocal of the fine-structure constant, 137.035 999 177 — famously near 137, and definitively not equal to it.

Rydberg constant measured

R_{\infty} = 10973731.568157\ \text{m}^{-1}

m⁻¹The wavenumber scale of atomic spectra, 10 973 731.568 157 m⁻¹ — the most precisely measured constant in all of physics.

Rydberg energy (hcR∞) measured

hcR_{\infty} = 2.179872361103 \times 10^{-18}\ \text{J}

JThe ionisation energy of ground-state hydrogen, 2.179 872 361 × 10⁻¹⁸ J or 13.605 693 eV — the natural unit of atomic energy.

Hartree energy measured

E_{\mathrm{h}} = 4.359744722206 \times 10^{-18}\ \text{J}

JThe atomic unit of energy, 4.359 744 722 × 10⁻¹⁸ J or 27.211 386 eV — twice the Rydberg and the currency of quantum chemistry.

Bohr radius measured

a_{0} = 5.29177210544 \times 10^{-11}\ \text{m}

mThe most probable electron-proton distance in ground-state hydrogen, 5.291 772 105 × 10⁻¹¹ m — the natural size of an atom.

Compton wavelength of the electron measured

\lambda_{\mathrm{C}} = 2.42631023538 \times 10^{-12}\ \text{m}

mλ_C = h/(m_e c) = 2.426 310 235 × 10⁻¹² m — the wavelength shift of a photon scattered through 90° by a free electron.

Reduced Compton wavelength of the electron measured

\bar{\lambda}_{\mathrm{C}} = 3.8615926744 \times 10^{-13}\ \text{m}

mħ/(m_e c) = 3.861 592 674 × 10⁻¹³ m — the Compton wavelength divided by 2π, and the natural length scale of the Dirac equation.

Classical electron radius measured

r_{\mathrm{e}} = 2.8179403205 \times 10^{-15}\ \text{m}

mr_e = α²a₀ = 2.817 940 321 × 10⁻¹⁵ m — the radius a classical sphere of charge e would need to have rest energy m_e c².

Thomson cross section measured

\sigma_{\mathrm{e}} = 6.6524587051 \times 10^{-29}\ \text{m}^{2}

m²The low-energy scattering cross section of a photon on a free electron, 6.652 458 705 × 10⁻²⁹ m² — that is 0.665 barn.

Electron g-factor measured

g_{\mathrm{e}} = -2.00231930436092

dimensionlessThe electron's magnetic moment in Bohr magnetons, −2.002 319 304 360 92 — the most precisely tested prediction in all of science.

Quantum of circulation measured

\frac{h}{2m_{\mathrm{e}}} = 0.00036369475467\ \text{m}^{2}\text{/s}

m²/sh/(2m_e) = 3.636 947 547 × 10⁻⁴ m²/s — the ratio of Planck's constant to mass that atom interferometers measure directly.

Electron charge-to-mass quotient measured

-e/m_{\mathrm{e}} = -1.75882000838 \times 10^{11}\ \text{C/kg}

C/kgThe electron's charge divided by its mass, −1.758 820 008 × 10¹¹ C/kg — the quantity J. J. Thomson measured in 1897 to discover the electron.

Proton charge-to-mass quotient measured

e/m_{\mathrm{p}} = 95788331.43\ \text{C/kg}

C/kg9.578 833 143 × 10⁷ C/kg — the proton's charge-to-mass ratio, smaller than the electron's by the full factor of 1836.

Proton rms charge radius measured

r_{\mathrm{p}} = 8.4075 \times 10^{-16}\ \text{m}

mThe root-mean-square radius of the proton's charge distribution, 8.4075 × 10⁻¹⁶ m — 0.841 femtometres, and recently controversial.

Nuclear radius constant

r_{0} = 1.2 \times 10^{-15}\ \text{m}

mThe empirical coefficient in R = r₀A^(1/3), about 1.2 × 10⁻¹⁵ m — the constant that says nuclear matter has a fixed density.

Barn (nuclear cross-section unit) exact

\mathrm{b} = 1 \times 10^{-28}\ \text{m}^{2}

m²Exactly 10⁻²⁸ m², or 100 fm²: the unit every nuclear cross section is quoted in, and roughly the geometric area of a uranium nucleus.

Planck length measured

\ell_{\mathrm{P}} = 1.616255 \times 10^{-35}\ \text{m}

m√(ħG/c³) = 1.616 255 × 10⁻³⁵ m — the length scale where quantum mechanics and gravity must both apply, and neither alone works.

Planck mass measured

m_{\mathrm{P}} = 2.176434 \times 10^{-8}\ \text{kg}

kg√(ħc/G) = 2.176 434 × 10⁻⁸ kg — about 22 micrograms, the only Planck unit on a human scale, and the mass where gravity meets quantum.

Planck time measured

t_{\mathrm{P}} = 5.391247 \times 10^{-44}\ \text{s}

s√(ħG/c⁵) = 5.391 247 × 10⁻⁴⁴ s — the time light takes to cross a Planck length, and the earliest instant physics can describe.

Planck temperature measured

T_{\mathrm{P}} = 1.416784 \times 10^{32}\ \text{K}

K√(ħc⁵/G)/k = 1.416 784 × 10³² K — the temperature at which thermal photons carry the Planck energy and gravity becomes quantum.

Planck energy measured

E_{\mathrm{P}} = 1,956,081,637\ \text{J}

Jm_P c² = 1.956 × 10⁹ J, or 1.221 × 10¹⁹ GeV — the energy scale of quantum gravity, and about the kinetic energy of a car on the motorway.

Fermi coupling constant measured

G_{\mathrm{F}}/(\hbar c)^{3} = 0.000011663787\ \text{GeV}^{-2}

GeV⁻²The strength of the weak interaction at low energy, 1.166 378 7 × 10⁻⁵ GeV⁻² — measured from the muon's 2.2 microsecond lifetime.

Electromagnetic 31

Vacuum magnetic permeability measured

\mu_{0} = 0.00000125663706127\ \text{N/A}^{2}\text{ (H/m)}

N/A² (H/m)How strongly a current magnetises empty space — the constant in Ampère's law, no longer exactly 4π×10⁻⁷ since the 2019 SI redefinition.

Vacuum electric permittivity measured

\varepsilon_{0} = 8.8541878188 \times 10^{-12}\ \text{F/m}

F/mThe electric constant of free space, setting the strength of Coulomb's law and the capacitance of every parallel-plate geometry.

Characteristic impedance of vacuum measured

Z_{0} = 376.730313412\ \text{Ω}

ΩThe ratio of electric to magnetic field strength in a plane wave in free space — the 377 ohms every antenna engineer matches to.

Coulomb constant measured

k_{e} = 8987551786.2\ \text{N}{\cdot}\text{m}^{2}\text{/C}^{2}

N·m²/C²The proportionality constant 1/(4πε₀) in Coulomb's law, fixing the enormous strength of the electrostatic force between charges.

Bohr magneton measured

\mu_{\mathrm{B}} = 9.2740100657 \times 10^{-24}\ \text{J/T}

J/TThe natural quantum of magnetic moment for an electron, eħ/2mₑ — the yardstick for atomic magnetism and electron spin.

Nuclear magneton measured

\mu_{\mathrm{N}} = 5.0507837393 \times 10^{-27}\ \text{J/T}

J/TThe magnetic-moment unit for nuclei, eħ/2mₚ — smaller than the Bohr magneton by the full proton-to-electron mass ratio of 1836.

Electron magnetic moment measured

\mu_{\mathrm{e}} = -9.2847646917 \times 10^{-24}\ \text{J/T}

J/TThe magnetic moment of a free electron, negative because its charge is, and about 0.116% larger than one Bohr magneton.

Proton magnetic moment measured

\mu_{\mathrm{p}} = 1.41060679545 \times 10^{-26}\ \text{J/T}

J/TThe magnetic moment of the proton, 2.79 nuclear magnetons rather than the 1 a point particle would show — evidence of quark structure.

Neutron magnetic moment measured

\mu_{\mathrm{n}} = -9.6623653 \times 10^{-27}\ \text{J/T}

J/TA neutral particle with a magnetic moment of −1.913 nuclear magnetons — proof on its own that the neutron has charged internal structure.

Magnetic flux quantum exact

\Phi_{0} = 2.067833848461929 \times 10^{-15}\ \text{Wb}

WbThe smallest unit of magnetic flux that can thread a superconducting loop, h/2e — exact since the 2019 SI redefinition fixed h and e.

Josephson constant exact

K_{\mathrm{J}} = 4.835978484169836 \times 10^{14}\ \text{Hz/V}

Hz/VThe frequency-to-voltage ratio 2e/h of a Josephson junction — 483.6 THz per volt, and the modern practical realisation of the volt.

von Klitzing constant exact

R_{\mathrm{K}} = 25812.8074593045\ \text{Ω}

ΩThe quantum Hall resistance h/e² ≈ 25.813 kΩ, exact since 2019 and the reference by which the ohm is now realised worldwide.

Conductance quantum exact

G_{0} = 0.00007748091729863649\ \text{S}

SThe conductance 2e²/h of a single ballistic quantum channel, about 77.5 μS — the step size in nanoscale wires and atomic point contacts.

Inverse conductance quantum exact

G_{0}^{-1} = 12906.40372965225\ \text{Ω}

ΩHalf the von Klitzing constant, h/2e² ≈ 12.906 kΩ — the resistance of one perfect ballistic channel and the floor for any nanoscale wire.

Faraday constant exact

F = 96485.33212331001\ \text{C/mol}

C/molThe charge carried by one mole of electrons, N_A×e ≈ 96485 coulombs — the bridge between the coulombs you meter and the moles you plate.

Proton gyromagnetic ratio measured

\gamma_{\mathrm{p}} = 267522187.08\ \text{s}^{-1}{\cdot}\text{T}^{-1}

s⁻¹·T⁻¹The proton's precession rate per unit magnetic field in angular frequency — the constant that turns a magnet strength into an NMR frequency.

Proton gyromagnetic ratio over 2π measured

\gamma_{\mathrm{p}}/2\pi = 42577478.461\ \text{Hz/T}

Hz/TThe proton Larmor frequency per tesla, 42.577 MHz/T — the number every NMR spectroscopist and MRI physicist works in directly.

Electron gyromagnetic ratio measured

\gamma_{\mathrm{e}} = 1.76085962784 \times 10^{11}\ \text{s}^{-1}{\cdot}\text{T}^{-1}

s⁻¹·T⁻¹The electron's spin precession rate per tesla, 658 times the proton's — the basis of electron spin resonance and of spin-qubit control.

Relative permittivity of air measured

\varepsilon_{r,\mathrm{air}} = 1.00059\ \text{—}

—Typical dielectric constant of dry air at 0 °C and one atmosphere — so close to vacuum that most capacitor and antenna work ignores the difference.

Relative permittivity of water measured

\varepsilon_{r,\mathrm{H_2O}} = 80.1\ \text{—}

—Typical static dielectric constant of liquid water at 20 °C — an outlier among common liquids and the reason water dissolves salts so well.

Relative permittivity of PTFE measured

\varepsilon_{r,\mathrm{PTFE}} = 2.1\ \text{—}

—Typical dielectric constant of PTFE (Teflon), about 2.1 and almost flat from DC to tens of gigahertz — the benchmark low-loss RF insulator.

Relative permittivity of FR-4 measured

\varepsilon_{r,\mathrm{FR\text{-}4}} = 4.4\ \text{—}

—Typical dielectric constant of FR-4 circuit-board laminate near 1 GHz — the number behind every microstrip impedance and trace-delay calculation.

Relative permeability of iron measured

\mu_{r,\mathrm{Fe}} = 5000\ \text{—}

—Typical maximum relative permeability of commercial soft iron — a wildly variable figure spanning roughly 200 to 5000 with purity and field level.

Relative permeability of mu-metal measured

\mu_{r,\mu\text{-metal}} = 80,000\ \text{—}

—Typical relative permeability of annealed mu-metal in weak fields — the nickel-iron alloy used to shield instruments from stray magnetic fields.

Speed of light in water measured

c_{\mathrm{H_2O}} = 224,900,000\ \text{m/s}

m/sLight travels through water at about 225,000 km/s, or c/1.333 — the slowing that bends a straw at the waterline and lets Cherenkov detectors work.

Speed of light in glass measured

c_{\mathrm{glass}} = 197,230,000\ \text{m/s}

m/sTypical propagation speed in crown glass, c/1.52 or about 197,000 km/s — the delay that makes lenses focus and optical fibres carry data.

Mains frequency (North America) measured

f_{60} = 60\ \text{Hz}

HzNominal 60 Hz AC power frequency across North America, held within about ±0.05 Hz by grid operators balancing generation against load.

Mains frequency (Europe and most of the world) measured

f_{50} = 50\ \text{Hz}

HzNominal 50 Hz AC power frequency used across Europe, Asia, Africa and Oceania, regulated to roughly ±0.05 Hz in normal grid operation.

Nominal mains voltage (North America) measured

V_{120} = 120\ \text{V}

VNominal 120 V RMS at North American outlets, with ANSI C84.1 allowing roughly 114–126 V at the point of utilisation under normal service.

Nominal mains voltage (Europe) measured

V_{230} = 230\ \text{V}

VNominal 230 V RMS single-phase supply under IEC 60038, the harmonised European figure that replaced the old 220 V and 240 V standards.

Earth's magnetic field strength measured

B_{\oplus} = 0.00005\ \text{T}

TTypical magnitude of the geomagnetic field at the surface, near 50 μT but ranging from about 25 μT at the equator to 65 μT near the poles.

Thermodynamic 37

Boltzmann constant exact

k_{\mathrm{B}} = 1.380649 \times 10^{-23}\ \text{J/K}

J/KThe energy per kelvin carried by a single particle's degree of freedom, fixed at exactly 1.380649e-23 J/K to define the kelvin.

Molar gas constant exact

R = 8.31446261815324\ \text{J/(mol}{\cdot}\text{K)}

J/(mol·K)The universal gas constant, R = k·N_A = 8.314462618 J/(mol·K), exact since 2019 and the R in PV = nRT and in every entropy table.

Stefan-Boltzmann constant exact

\sigma = 5.670374419 \times 10^{-8}\ \text{W/(m}^{2}{\cdot}\text{K}^{4}\text{)}

W/(m²·K⁴)Radiant emittance of a blackbody per fourth power of temperature: 5.670374419e-8 W/(m²·K⁴), exact in the post-2019 SI.

Wien displacement law constant (wavelength) exact

b = 0.002897771955\ \text{m}{\cdot}\text{K}

m·KWien's constant b = 2.897771955e-3 m·K: divide by absolute temperature to get the wavelength where a blackbody's spectrum peaks.

Wien displacement law constant (frequency) exact

b' = 5.878925757 \times 10^{10}\ \text{Hz/K}

Hz/KFrequency form of Wien's law, b' = 5.878925757e10 Hz/K: the peak frequency of a blackbody is b'·T, not c divided by the peak wavelength.

First radiation constant exact

c_1 = 3.741771852 \times 10^{-16}\ \text{W}{\cdot}\text{m}^{2}

W·m²c₁ = 2πhc² = 3.741771852e-16 W·m², the numerator of Planck's law in its spectral exitance form and the scale of all blackbody emission.

Second radiation constant exact

c_2 = 0.01438776877\ \text{m}{\cdot}\text{K}

m·Kc₂ = hc/k = 1.438776877e-2 m·K, the constant in the exponent of Planck's law and the basis of radiation thermometry.

Molar volume of an ideal gas at STP (0 °C, 100 kPa) exact

V_{\mathrm{m}}^{\,\mathrm{STP}} = 22.71095464\ \text{L/mol}

L/mol22.71095464 L/mol at IUPAC standard temperature and pressure, 273.15 K and 100 kPa exactly — not the older 22.4 L/mol.

Molar volume of an ideal gas at 0 °C and 1 atm exact

V_{\mathrm{m}}^{\,\mathrm{atm}} = 22.41396954\ \text{L/mol}

L/molThe textbook 22.414 L/mol: one mole of ideal gas at 273.15 K and 101.325 kPa, the pre-1982 definition of standard conditions.

Molar volume of an ideal gas at 25 °C and 1 atm exact

V_{\mathrm{m}}^{\,\mathrm{NTP}} = 24.4654037\ \text{L/mol}

L/mol24.4654 L/mol at 298.15 K and 101.325 kPa, the ambient reference used for gas concentrations in ppm-to-mg/m³ conversions.

Loschmidt constant exact

n_0 = 2.686780111 \times 10^{25}\ \text{m}^{-3}

m⁻³Number density of an ideal gas at 273.15 K and 101.325 kPa: 2.6867801e25 molecules per cubic metre, or 2.69e19 per cubic centimetre.

Sackur-Tetrode constant measured

S_0/R = -1.15170754\ \text{—}

—Reduced absolute entropy of an ideal monatomic gas at 1 K and 100 kPa, -1.1517075, the constant that puts Planck's h inside a classical gas.

Molar mass constant measured

M_{\mathrm{u}} = 0.00100000000105\ \text{kg/mol}

kg/molM_u = 1.00000000105e-3 kg/mol, the factor turning a relative atomic mass into a molar mass — no longer exactly 1 g/mol since 2019.

Standard atmosphere exact

p_{\mathrm{atm}} = 101,325\ \text{Pa}

PaOne standard atmosphere is exactly 101325 Pa, equal to 14.6959 psi, 760 mmHg, 29.921 inHg or 1.01325 bar, by international definition.

Standard state pressure exact

p^{\circ} = 100,000\ \text{Pa}

PaThe thermodynamic standard state pressure, exactly 1 bar = 100 kPa, the p° in every tabulated ΔG°, ΔH° and equilibrium constant.

Absolute zero exact

T = 0\ \mathrm{K} = 0\ \text{K}

KThe zero of the thermodynamic temperature scale: 0 K, equal to -273.15 °C and -459.67 °F exactly, both figures fixed by definition.

Ice point (0 °C in kelvin) exact

T_0 = 273.15\ \text{K}

KZero degrees Celsius is exactly 273.15 K, the offset that converts every Celsius reading to absolute temperature in gas-law work.

Triple point of water measured

T_{\mathrm{TPW}} = 273.16\ \text{K}

KThe unique 273.16 K (0.01 °C) at which ice, liquid water and vapour coexist — exact until 2019, now a measured value good to 0.1 mK.

Triple point pressure of water measured

p_{\mathrm{TPW}} = 611.657\ \text{Pa}

Pa611.657 Pa, about 0.6 % of an atmosphere: the vapour pressure at water's triple point and the floor below which liquid water cannot exist.

Normal boiling point of water

T_{\mathrm{b}} = 373.1243\ \text{K}

KWater boils at 99.9743 °C (373.1243 K) under one standard atmosphere — very slightly below 100 °C, and not by accident.

Specific gas constant for dry air

R_{\mathrm{air}} = 287.0528\ \text{J/(kg}{\cdot}\text{K)}

J/(kg·K)R/M for dry air, 287.0528 J/(kg·K), using the standard-atmosphere molar mass 28.9644 g/mol — the R in p = ρRT for air.

Specific gas constant for water vapour

R_{\mathrm{v}} = 461.523\ \text{J/(kg}{\cdot}\text{K)}

J/(kg·K)R/M for water vapour, 461.523 J/(kg·K), from a molar mass of 18.015268 g/mol — the constant behind psychrometrics and humidity ratio.

Heat capacity ratio of air

\gamma_{\mathrm{air}} = 1.4\ \text{—}

—γ = cp/cv = 1.400 for dry air near 20 °C and 1 atm, the exponent in adiabatic compression and in the speed of sound.

Heat capacity ratio of argon

\gamma_{\mathrm{Ar}} = 1.667\ \text{—}

—γ = 1.667 for argon and the other monatomic gases, the theoretical maximum 5/3 predicted by kinetic theory for point-like atoms.

Heat capacity ratio of steam

\gamma_{\mathrm{steam}} = 1.33\ \text{—}

—γ ≈ 1.33 for low-pressure steam at 100 °C: a triatomic bent molecule with rotational modes that lower it well below air's 1.40.

Specific heat capacity of liquid water

c_{p,\mathrm{water}} = 4181.6\ \text{J/(kg}{\cdot}\text{K)}

J/(kg·K)4181.6 J/(kg·K) for liquid water at 25 °C and 0.1 MPa — about 1.00 BTU/(lb·°F), the highest of any common liquid.

Specific heat capacity of ice

c_{p,\mathrm{ice}} = 2108\ \text{J/(kg}{\cdot}\text{K)}

J/(kg·K)About 2108 J/(kg·K) for ice at 0 °C, roughly half the value for liquid water — the reason freezers cool loads far faster than they freeze them.

Specific heat capacity of dry air

c_{p,\mathrm{air}} = 1005\ \text{J/(kg}{\cdot}\text{K)}

J/(kg·K)About 1005 J/(kg·K) at constant pressure for dry air near 300 K and 1 atm; cv is 718 J/(kg·K), and their ratio is γ = 1.40.

Latent heat of fusion of water

L_{\mathrm{f}} = 333,550\ \text{J/kg}

J/kg333.55 kJ/kg (143.4 BTU/lb) to melt ice at 0 °C without changing its temperature — equivalent to 80 K of sensible heating of water.

Latent heat of vaporisation of water

L_{\mathrm{v}} = 2,256,400\ \text{J/kg}

J/kg2256.4 kJ/kg (970 BTU/lb) to boil water at 100 °C and 1 atm, nearly seven times the heat of fusion and the basis of all steam heating.

Maximum density of water (4 °C) measured

\rho_{\max} = 999.975\ \text{kg/m}^{3}

kg/m³Water is densest at about 3.98 °C, 999.975 kg/m³ — the anomaly that makes ice float and keeps deep lakes from freezing solid.

Density of water at 20 °C measured

\rho_{20} = 998.207\ \text{kg/m}^{3}

kg/m³998.207 kg/m³ at 20 °C and 1 atm (62.316 lb/ft³), the reference density behind specific gravity and most laboratory calibrations.

Mechanical equivalent of heat exact

J = 4.1868\ \text{J/cal (IT)}

J/cal (IT)4.1868 joules per international-table calorie, exact by definition — the conversion Joule spent two decades measuring by hand.

Calorie (thermochemical) exact

\mathrm{cal} = 4.184\ \text{J}

JThe thermochemical calorie is exactly 4.184 J; the food Calorie is a kilocalorie, 4184 J, a factor of a thousand larger.

British thermal unit exact

\mathrm{BTU} = 1055.05585262\ \text{J}

JThe international-table BTU is exactly 1055.05585262 J, the heat that raises one pound of water by one degree Fahrenheit.

Ton of refrigeration exact

\mathrm{TR} = 3516.8528420667\ \text{W}

WExactly 12000 BTU/h = 3516.85 W: the cooling rate that melts one short ton of ice in 24 hours, still the unit chillers are sold in.

Boiler horsepower

\mathrm{bhp} = 9809.5\ \text{W}

W9809.5 W (33475 BTU/h): the heat rate to evaporate 34.5 lb/h of water at 212 °F, and nothing at all to do with mechanical horsepower.

Astronomical 47

Astronomical Unit exact

\mathrm{au} = 1.495978707 \times 10^{11}\ \text{m}

mThe Sun–Earth yardstick, fixed by the IAU in 2012 as exactly 149 597 870 700 m and no longer tied to Earth's actual orbit.

Light-Year exact

\mathrm{ly} = 9.4607304725808 \times 10^{15}\ \text{m}

mDistance light travels in one Julian year of 365.25 days — exactly 9 460 730 472 580 800 m, since both c and the year are defined.

Parsec exact

\mathrm{pc} = 3.085677581491367 \times 10^{16}\ \text{m}

mDistance at which one au subtends one arcsecond — 648000/π au exactly, about 3.26 light-years, the working unit of stellar astronomy.

Solar Mass measured

M_\odot = 1.98841 \times 10^{30}\ \text{kg}

kgMass of the Sun, about 333 000 Earths and 99.86 per cent of all matter in the solar system — the yardstick for every stellar mass.

Nominal Solar Radius exact

\mathcal{R}^{\mathrm{N}}_\odot = 695,700,000\ \text{m}

mIAU nominal solar radius, exactly 6.957 × 10⁸ m — a fixed convention, since a gaseous Sun has no true surface to measure.

Nominal Solar Luminosity exact

\mathcal{L}^{\mathrm{N}}_\odot = 3.828 \times 10^{26}\ \text{W}

WIAU nominal solar luminosity, exactly 3.828 × 10²⁶ W — the conventional unit in which every other star's power output is quoted.

Solar Effective Temperature exact

\mathcal{T}^{\mathrm{N}}_{\mathrm{eff}\odot} = 5772\ \text{K}

KIAU nominal effective temperature of the Sun, 5772 K — the blackbody temperature that radiates the solar luminosity from the solar radius.

Solar Constant (Total Solar Irradiance) measured

S_0 = 1361\ \text{W/m}^{2}

W/m²Total solar irradiance above the atmosphere at one au, about 1361 W/m² — the input to every climate model and solar panel estimate.

Standard Gravitational Parameter of the Sun measured

GM_\odot = 1.32712440041 \times 10^{20}\ \text{m}^{3}\text{/s}^{2}

m³/s²The heliocentric gravitational constant GM⊙, known to ten digits from planetary radar — far better than the Sun's mass in kilograms.

Schwarzschild Radius of the Sun measured

r_{s,\odot} = 2953.25\ \text{m}

mRadius to which the Sun would have to be crushed to become a black hole, 2GM⊙/c² — a shade under three kilometres.

Solar Wind Speed (typical)

v_{\mathrm{sw}} = 400,000\ \text{m/s}

m/sTypical speed of the solar wind at Earth's orbit, about 400 km/s; the slow and fast streams range from roughly 300 to 800 km/s.

Mass of the Earth measured

M_\oplus = 5.9722 \times 10^{24}\ \text{kg}

kgMass of the Earth, 5.9722 × 10²⁴ kg — the unit in which rocky exoplanets are weighed, and limited in precision only by G.

Earth Equatorial Radius exact

a_\oplus = 6,378,137\ \text{m}

mSemi-major axis of the WGS 84 reference ellipsoid, exactly 6 378 137 m — the equatorial radius every GPS receiver is built around.

Earth Polar Radius exact

b_\oplus = 6356752.314245\ \text{m}

mSemi-minor axis of the WGS 84 ellipsoid, 6 356 752.3 m — 21.4 km shorter than the equatorial radius because the Earth is spinning.

Earth Mean Radius

R_\oplus = 6371008.8\ \text{m}

mMean radius (2a + b)/3 of the WGS 84 ellipsoid, about 6371 km — the single figure used when a spherical Earth is good enough.

Standard Gravitational Parameter of the Earth measured

GM_\oplus = 3.986004418 \times 10^{14}\ \text{m}^{3}\text{/s}^{2}

m³/s²The geocentric gravitational constant GM⊕, 3.986 004 418 × 10¹⁴ m³/s², known to nine digits and used by every GPS satellite.

Earth Mean Orbital Speed

v_\oplus = 29,780\ \text{m/s}

m/sMean speed of the Earth along its orbit, about 29.78 km/s — roughly 107 000 km/h, and it varies with distance from the Sun.

Earth Orbit Semi-Major Axis

a_{\oplus,\mathrm{orb}} = 1.495982612 \times 10^{11}\ \text{m}

mEarth's actual mean orbital distance, 1.000 002 61 au — close to the astronomical unit but a measured quantity, not the definition.

Earth Orbital Eccentricity

e_\oplus = 0.0167086\ \text{—}

—Eccentricity of Earth's orbit at J2000, 0.0167 — nearly circular, yet enough to vary sunlight at the top of the atmosphere by 6.8 per cent.

Earth Axial Tilt (Obliquity of the Ecliptic)

\varepsilon = 0.4090928\ \text{rad}

radObliquity of the ecliptic at J2000, 23.4393° or 0.409 rad — the tilt of Earth's spin axis that causes the seasons.

Escape Velocity of the Earth

v_{\mathrm{esc},\oplus} = 11,186\ \text{m/s}

m/sSpeed needed to break free of Earth's gravity from the surface, about 11.19 km/s, ignoring atmospheric drag and the planet's rotation.

Sidereal Day

T_{\mathrm{sid}} = 86164.0905\ \text{s}

sEarth's rotation period relative to the fixed stars, 23 h 56 min 4.09 s — about four minutes shorter than the solar day.

Mean Solar Day exact

d = 86,400\ \text{s}

sThe civil day of exactly 86 400 SI seconds — a defined unit that Earth's actual rotation now overruns by a millisecond or two.

Julian Year exact

a_{\mathrm{J}} = 31,557,600\ \text{s}

sExactly 365.25 days of 86 400 SI seconds — the conventional astronomical year that defines the light-year and the Julian century.

Sidereal Year

T_{\mathrm{sid}}^{\mathrm{yr}} = 31558149.8\ \text{s}

sOne orbit of the Sun relative to the fixed stars, 365.2564 days — about 20 minutes longer than the tropical year of the seasons.

Tropical Year

T_{\mathrm{trop}} = 31556925.2\ \text{s}

sEquinox to equinox, 365.2422 days — the year the seasons follow, and the quantity every calendar reform has tried to approximate.

Mass of the Moon measured

M_{\mathrm{Moon}} = 7.3459 \times 10^{22}\ \text{kg}

kgMass of the Moon, 7.346 × 10²² kg — 1.23 per cent of Earth's, the largest satellite-to-planet mass ratio in the solar system.

Mean Radius of the Moon measured

R_{\mathrm{Moon}} = 1,737,400\ \text{m}

mVolumetric mean radius of the Moon, 1737.4 km — just over a quarter of Earth's radius, and barely 0.3 per cent from a perfect sphere.

Mean Earth–Moon Distance

a_{\mathrm{Moon}} = 384,400,000\ \text{m}

mSemi-major axis of the lunar orbit, 384 400 km centre to centre; the actual distance ranges from 356 500 to 406 700 km.

Surface Gravity of the Moon measured

g_{\mathrm{Moon}} = 1.62\ \text{m/s}^{2}

m/s²Gravitational acceleration at the lunar surface, 1.62 m/s² — one sixth of Earth's, the value the Apollo crews had to learn to walk in.

Escape Velocity of the Moon

v_{\mathrm{esc},\mathrm{Moon}} = 2380\ \text{m/s}

m/sSpeed needed to leave the Moon's gravity from its surface, about 2.38 km/s — roughly a fifth of Earth's escape velocity.

Mass of Mars measured

M_{\mathrm{Mars}} = 6.4171 \times 10^{23}\ \text{kg}

kgMass of Mars, 6.417 × 10²³ kg — about 10.7 per cent of Earth's, small enough that the planet lost most of its atmosphere.

Mean Radius of Mars measured

R_{\mathrm{Mars}} = 3,389,500\ \text{m}

mVolumetric mean radius of Mars, 3389.5 km; the equatorial radius is 3396.2 km and the polar 3376.2 km, a 20 km flattening.

Surface Gravity of Mars measured

g_{\mathrm{Mars}} = 3.71\ \text{m/s}^{2}

m/s²Equatorial surface gravity on Mars, 3.71 m/s² — 38 per cent of Earth's, the figure every Mars lander design is built around.

Escape Velocity of Mars

v_{\mathrm{esc},\mathrm{Mars}} = 5030\ \text{m/s}

m/sSpeed needed to escape Mars from the surface, about 5.03 km/s — less than half Earth's, which is why a return mission is even thinkable.

Mass of Mercury measured

M_{\mathrm{Mercury}} = 3.3011 \times 10^{23}\ \text{kg}

kgMass of Mercury, 3.301 × 10²³ kg — the smallest planet, yet the second densest, with an iron core filling most of its volume.

Mass of Venus measured

M_{\mathrm{Venus}} = 4.8675 \times 10^{24}\ \text{kg}

kgMass of Venus, 4.8675 × 10²⁴ kg — 81.5 per cent of Earth's, making it our closest twin in bulk and nothing like it in climate.

Mass of Jupiter measured

M_{\mathrm{J}} = 1.8982 \times 10^{27}\ \text{kg}

kgMass of Jupiter, 1.898 × 10²⁷ kg — 318 Earths, and more than twice all the other planets combined; the unit for weighing exoplanets.

Equatorial Radius of Jupiter measured

R_{\mathrm{J}} = 71,492,000\ \text{m}

mJupiter's equatorial radius at the 1-bar level, 71 492 km; the polar radius is 66 854 km, a 6.5 per cent flattening from fast rotation.

Mass of Saturn measured

M_{\mathrm{Saturn}} = 5.6834 \times 10^{26}\ \text{kg}

kgMass of Saturn, 5.683 × 10²⁶ kg — 95 Earths spread so thinly that its mean density, 687 kg/m³, is less than that of water.

Mass of Uranus measured

M_{\mathrm{Uranus}} = 8.681 \times 10^{25}\ \text{kg}

kgMass of Uranus, 8.681 × 10²⁵ kg — 14.5 Earths of hydrogen, helium and icy volatiles, tipped on its side at 98 degrees.

Mass of Neptune measured

M_{\mathrm{Neptune}} = 1.02413 \times 10^{26}\ \text{kg}

kgMass of Neptune, 1.024 × 10²⁶ kg — 17.1 Earths, the densest of the giant planets and the one found with mathematics before a telescope.

Hubble Constant measured

H_0 = 67.4\ \text{km/(s}{\cdot}\text{Mpc)}

km/(s·Mpc)Present expansion rate of the universe, 67.4 km/(s·Mpc) from the cosmic microwave background — a galaxy 1 Mpc away recedes at 67 km/s.

Age of the Universe measured

t_0 = 4.354 \times 10^{17}\ \text{s}

sTime since the Big Bang, 13.797 billion years or 4.35 × 10¹⁷ seconds, from fitting the ΛCDM model to the microwave background.

Critical Density of the Universe measured

\rho_c = 8.53 \times 10^{-27}\ \text{kg/m}^{3}

kg/m³Density 3H₀²/8πG that makes the universe spatially flat, about 8.5 × 10⁻²⁷ kg/m³ — some five hydrogen atoms per cubic metre.

Cosmic Microwave Background Temperature measured

T_{\mathrm{CMB}} = 2.72548\ \text{K}

KTemperature of the relic radiation from the Big Bang, 2.725 48 K — the most perfect blackbody spectrum ever measured, anywhere.

Chandrasekhar Limit measured

M_{\mathrm{Ch}} = 2.86 \times 10^{30}\ \text{kg}

kgMaximum mass a white dwarf can support by electron degeneracy pressure, about 1.44 solar masses or 2.86 × 10³⁰ kg.

Material Properties 69

Density of Seawater measured

\rho_{\mathrm{sw}} = 1025\ \text{kg/m}^{3}

kg/m³Representative density of open-ocean seawater at 35 g/kg salinity and 15 °C, about 1025 kg/m³ or 64 lb/ft³ at the surface.

Density of Dry Air at 20 °C measured

\rho_{\mathrm{air}} = 1.204\ \text{kg/m}^{3}

kg/m³Density of dry air at 20 °C and 101.325 kPa, 1.204 kg/m³ — the standard-air value behind the 1.08 sensible-heat factor.

Density of Structural Steel measured

\rho_{\mathrm{steel}} = 7850\ \text{kg/m}^{3}

kg/m³Typical density of carbon and low-alloy structural steel at 20 °C, 7850 kg/m³ or 490 lb/ft³, essentially independent of grade.

Density of Aluminium Alloy 6061 measured

\rho_{\mathrm{Al}} = 2700\ \text{kg/m}^{3}

kg/m³Typical density of 6061 aluminium alloy at 20 °C, 2700 kg/m³ or 169 lb/ft³ — about 35 % of steel for the same volume.

Density of Reinforced Concrete measured

\rho_{\mathrm{RC}} = 2400\ \text{kg/m}^{3}

kg/m³Typical density of normal-weight reinforced concrete, 2400 kg/m³ or 150 lb/ft³, including ordinary reinforcing steel content.

Density of Ice at 0 °C measured

\rho_{\mathrm{ice}} = 917\ \text{kg/m}^{3}

kg/m³Density of ordinary hexagonal ice at 0 °C and 1 atm, about 917 kg/m³ — roughly 8 % lighter than the water it freezes from.

Density of Mercury at 20 °C measured

\rho_{\mathrm{Hg}} = 13,534\ \text{kg/m}^{3}

kg/m³Density of liquid mercury at 20 °C and 1 atm, 13 534 kg/m³ — 13.5 times water, and the basis of the mmHg pressure unit.

Density of Softwood Timber measured

\rho_{\mathrm{wood}} = 500\ \text{kg/m}^{3}

kg/m³Representative density of construction softwood such as spruce-pine-fir or Douglas fir at 12 % moisture, roughly 500 kg/m³.

Young's Modulus of Structural Steel measured

E_{\mathrm{steel}} = 2 \times 10^{11}\ \text{Pa}

PaElastic modulus of carbon and low-alloy structural steel at room temperature, 200 GPa or 29 000 ksi, effectively grade-independent.

Young's Modulus of Type 304 Stainless Steel measured

E_{\mathrm{SS304}} = 1.93 \times 10^{11}\ \text{Pa}

PaElastic modulus of annealed Type 304 austenitic stainless steel at 20 °C, about 193 GPa or 28 000 ksi — 3 % below carbon steel.

Young's Modulus of Aluminium Alloy 6061 measured

E_{\mathrm{Al}} = 6.89 \times 10^{10}\ \text{Pa}

PaElastic modulus of 6061 aluminium at room temperature, 68.9 GPa or 10 000 ksi — roughly one third the stiffness of steel.

Young's Modulus of Normal-Weight Concrete measured

E_{c} = 2.5 \times 10^{10}\ \text{Pa}

PaSecant elastic modulus of 28 MPa (4000 psi) normal-weight concrete, about 25 GPa — computed from strength, not measured directly.

Shear Modulus of Structural Steel measured

G_{\mathrm{steel}} = 7.72 \times 10^{10}\ \text{Pa}

PaShear (rigidity) modulus of structural steel at room temperature, 77.2 GPa or 11 200 ksi — the value used in torsion and shear.

Shear Modulus of Aluminium Alloy measured

G_{\mathrm{Al}} = 2.6 \times 10^{10}\ \text{Pa}

PaShear modulus of common wrought aluminium alloys at room temperature, about 26 GPa or 3800 ksi — one third of steel's value.

Poisson's Ratio of Steel measured

\nu_{\mathrm{steel}} = 0.3\ \text{—}

—Poisson's ratio of carbon and alloy steel in the elastic range, 0.30 — the lateral contraction per unit of axial extension.

Poisson's Ratio of Aluminium measured

\nu_{\mathrm{Al}} = 0.33\ \text{—}

—Poisson's ratio of wrought aluminium alloys in the elastic range, about 0.33 — slightly higher than steel's 0.30.

Poisson's Ratio of Concrete measured

\nu_{c} = 0.2\ \text{—}

—Poisson's ratio of hardened normal-weight concrete under service compression, typically 0.15–0.25 with 0.20 used in design.

Poisson's Ratio of Rubber measured

\nu_{\mathrm{rubber}} = 0.499\ \text{—}

—Poisson's ratio of natural and synthetic rubber, about 0.499 — nearly incompressible, the practical limit for isotropic solids.

Yield Strength of ASTM A36 Steel

F_{y,\mathrm{A36}} = 248,000,000\ \text{Pa}

PaSpecified minimum yield strength of ASTM A36 structural steel, 36 ksi or 248 MPa — a floor guaranteed by the mill, not a measurement.

Yield Strength of ASTM A992 Steel

F_{y,\mathrm{A992}} = 345,000,000\ \text{Pa}

PaSpecified minimum yield strength of ASTM A992 wide-flange steel, 50 ksi or 345 MPa — the default grade for W-shapes since 1998.

Yield Strength of 6061-T6 Aluminium measured

F_{y,\mathrm{6061\text{-}T6}} = 276,000,000\ \text{Pa}

PaTypical 0.2 % offset yield strength of 6061-T6 aluminium, 276 MPa or 40 ksi, with an ultimate tensile strength near 310 MPa.

Compressive Strength of Normal-Weight Concrete measured

f'_{c} = 28,000,000\ \text{Pa}

PaTypical specified 28-day cylinder strength of ordinary structural concrete, about 28 MPa (4000 psi), with 20–40 MPa the usual range.

Tensile Strength of a Grade 5 Bolt

S_{ut,\mathrm{Gr.5}} = 827,000,000\ \text{Pa}

PaMinimum ultimate tensile strength of an SAE Grade 5 bolt up to 1 in diameter, 120 ksi or 827 MPa, with 85 ksi proof strength.

Tensile Strength of a Grade 8 Bolt

S_{ut,\mathrm{Gr.8}} = 1,034,000,000\ \text{Pa}

PaMinimum ultimate tensile strength of an SAE Grade 8 bolt, 150 ksi or 1034 MPa, with 130 ksi proof — the high-strength shop fastener.

Thermal Conductivity of Copper measured

k_{\mathrm{Cu}} = 401\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Thermal conductivity of pure annealed copper at 25 °C, about 401 W/(m·K) — the benchmark for practical heat-transfer materials.

Thermal Conductivity of Aluminium measured

k_{\mathrm{Al}} = 237\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Thermal conductivity of pure aluminium at 25 °C, about 237 W/(m·K); alloy 6061-T6 is markedly lower at roughly 167 W/(m·K).

Thermal Conductivity of Carbon Steel measured

k_{\mathrm{steel}} = 50\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Thermal conductivity of plain carbon steel near room temperature, roughly 50 W/(m·K) — about an eighth of copper's value.

Thermal Conductivity of Type 304 Stainless Steel measured

k_{\mathrm{SS304}} = 16.2\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Thermal conductivity of Type 304 austenitic stainless steel at 20 °C, about 16 W/(m·K) — roughly a third of carbon steel's.

Thermal Conductivity of Water measured

k_{w} = 0.598\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Thermal conductivity of liquid water at 20 °C and 1 atm, about 0.60 W/(m·K) — high for a liquid, still 700× worse than copper.

Thermal Conductivity of Air measured

k_{\mathrm{air}} = 0.0257\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Thermal conductivity of dry air at 20 °C and 1 atm, about 0.026 W/(m·K) — the benchmark every insulation is measured against.

Thermal Conductivity of Fibreglass Batt Insulation measured

k_{\mathrm{fg}} = 0.04\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Thermal conductivity of standard fibreglass batt at 24 °C, about 0.040 W/(m·K) — roughly R-3.6 per inch in US units.

Thermal Conductivity of Rigid Foam Board measured

k_{\mathrm{foam}} = 0.024\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Aged thermal conductivity of rigid polyisocyanurate or XPS board, roughly 0.024–0.029 W/(m·K), about R-5 to R-6 per inch.

Thermal Conductivity of Concrete measured

k_{c} = 1.7\ \text{W/(m}{\cdot}\text{K)}

W/(m·K)Thermal conductivity of normal-weight structural concrete, roughly 1.4–2.0 W/(m·K) depending on aggregate and moisture content.

Specific Heat of Carbon Steel measured

c_{\mathrm{steel}} = 486\ \text{J/(kg}{\cdot}\text{K)}

J/(kg·K)Specific heat of plain carbon steel near room temperature, about 486 J/(kg·K) or 0.116 BTU/(lb·°F), rising with temperature.

Specific Heat of 30 % Propylene Glycol measured

c_{\mathrm{PG30}} = 3850\ \text{J/(kg}{\cdot}\text{K)}

J/(kg·K)Specific heat of a 30 % by volume propylene glycol/water mix near 40 °C, about 3850 J/(kg·K) — some 8 % below plain water.

Thermal Expansion Coefficient of Carbon Steel measured

\alpha_{\mathrm{steel}} = 0.0000117\ \text{1/K}

1/KLinear thermal expansion coefficient of carbon steel near room temperature, 11.7 µm/(m·K) or 6.5 µin/(in·°F).

Thermal Expansion Coefficient of Type 304 Stainless measured

\alpha_{\mathrm{SS304}} = 0.0000173\ \text{1/K}

1/KLinear expansion coefficient of Type 304 austenitic stainless steel, 17.3 µm/(m·K) — about 50 % more than carbon steel.

Thermal Expansion Coefficient of Aluminium measured

\alpha_{\mathrm{Al}} = 0.0000234\ \text{1/K}

1/KLinear expansion coefficient of aluminium near room temperature, 23.4 µm/(m·K) or 13 µin/(in·°F) — twice that of steel.

Thermal Expansion Coefficient of Copper measured

\alpha_{\mathrm{Cu}} = 0.0000168\ \text{1/K}

1/KLinear expansion coefficient of copper near room temperature, 16.8 µm/(m·K) or 9.3 µin/(in·°F) — 44 % more than steel.

Thermal Expansion Coefficient of Concrete measured

\alpha_{c} = 0.00001\ \text{1/K}

1/KLinear expansion coefficient of normal-weight concrete, roughly 8–12 µm/(m·K) — close enough to steel to make reinforcing work.

Thermal Expansion Coefficient of PVC measured

\alpha_{\mathrm{PVC}} = 0.000054\ \text{1/K}

1/KLinear expansion coefficient of rigid PVC pipe, about 54 µm/(m·K) or 3.0 µin/(in·°F) — nearly five times that of steel.

Dynamic Viscosity of Water at 20 °C measured

\mu_{w} = 0.001002\ \text{Pa}{\cdot}\text{s}

Pa·sDynamic viscosity of pure water at 20 °C and 1 atm, 1.002 mPa·s — the value that made the centipoise a de facto standard.

Dynamic Viscosity of Air at 20 °C measured

\mu_{\mathrm{air}} = 0.0000181\ \text{Pa}{\cdot}\text{s}

Pa·sDynamic viscosity of dry air at 20 °C and 1 atm, 18.1 µPa·s — about 1/55 of water's, though its kinematic viscosity is 15× larger.

Dynamic Viscosity of SAE 30 Oil measured

\mu_{\mathrm{SAE30}} = 0.088\ \text{Pa}{\cdot}\text{s}

Pa·sDynamic viscosity of a typical SAE 30 mineral engine oil at 40 °C, roughly 0.088 Pa·s — about 90 times that of water at 20 °C.

Electrical Resistivity of Copper at 20 °C measured

\rho_{\mathrm{Cu}} = 1.678 \times 10^{-8}\ \text{Ω}{\cdot}\text{m}

Ω·mResistivity of pure annealed copper at 20 °C, 1.678 × 10⁻⁸ Ω·m; the commercial IACS reference is 1.7241 × 10⁻⁸ Ω·m.

Electrical Resistivity of Aluminium at 20 °C measured

\rho_{\mathrm{Al}} = 2.65 \times 10^{-8}\ \text{Ω}{\cdot}\text{m}

Ω·mResistivity of pure aluminium at 20 °C, 2.65 × 10⁻⁸ Ω·m; EC-grade 1350 conductor is about 2.83 × 10⁻⁸ Ω·m (61 % IACS).

Electrical Resistivity of Carbon Steel measured

\rho_{\mathrm{steel}} = 1.6 \times 10^{-7}\ \text{Ω}{\cdot}\text{m}

Ω·mResistivity of plain carbon steel at 20 °C, roughly 1.4–1.8 × 10⁻⁷ Ω·m — about ten times copper's, and composition-sensitive.

Temperature Coefficient of Resistance, Copper measured

\alpha_{\mathrm{Cu}} = 0.00393\ \text{1/K}

1/KTemperature coefficient of resistance for annealed copper referenced to 20 °C, 0.00393 per kelvin — 0.393 % more resistance per degree.

Speed of Sound in Dry Air at 20 °C measured

c_{\mathrm{air}} = 343.2\ \text{m/s}

m/sSpeed of sound in dry air at 20 °C and 1 atm, 343 m/s or 1125 ft/s — set by temperature, essentially not by pressure.

Speed of Sound in Water at 20 °C measured

c_{w} = 1482\ \text{m/s}

m/sSpeed of sound in fresh water at 20 °C and 1 atm, about 1482 m/s — 4.3 times faster than in air, and rising with temperature.

Speed of Sound in Steel measured

c_{\mathrm{steel}} = 5900\ \text{m/s}

m/sLongitudinal (bulk) wave speed in carbon steel, about 5900 m/s; the thin-bar wave speed √(E/ρ) is lower, near 5100 m/s.

Friction Coefficient, Dry Steel on Steel measured

\mu_{\mathrm{steel/steel}} = 0.6\ \text{—}

—Representative static friction coefficient for clean dry steel on steel, about 0.6 with a legitimate range of 0.4 to 0.8.

Friction Coefficient, Lubricated Steel on Steel measured

\mu_{\mathrm{steel/steel,lub}} = 0.1\ \text{—}

—Representative friction coefficient for oil-lubricated steel on steel in boundary lubrication, about 0.10 (range 0.05–0.15).

Friction Coefficient, Rubber Tyre on Dry Asphalt measured

\mu_{\mathrm{dry}} = 0.8\ \text{—}

—Representative peak friction coefficient for a passenger tyre on dry asphalt, about 0.8 — competition tyres exceed 1.0.

Friction Coefficient, Rubber Tyre on Wet Asphalt measured

\mu_{\mathrm{wet}} = 0.5\ \text{—}

—Representative friction coefficient for a passenger tyre on wet asphalt, about 0.5, falling to 0.3 or below on worn tyres.

Emissivity of Polished Aluminium measured

\varepsilon_{\mathrm{Al}} = 0.05\ \text{—}

—Total hemispherical emissivity of bright polished aluminium near room temperature, about 0.05 — an excellent radiant barrier.

Emissivity of Oxidised Steel measured

\varepsilon_{\mathrm{steel}} = 0.8\ \text{—}

—Total emissivity of oxidised or mill-scaled carbon steel, about 0.8 — against roughly 0.1 for the same steel polished bright.

Emissivity of Flat Black Paint measured

\varepsilon_{\mathrm{black}} = 0.96\ \text{—}

—Total hemispherical emissivity of flat black paint near room temperature, about 0.96 — the practical stand-in for a black body.

Absolute Roughness of Commercial Steel Pipe measured

\epsilon_{\mathrm{steel}} = 0.000046\ \text{m}

mAbsolute roughness ε of new commercial steel or wrought-iron pipe, 0.046 mm (0.00015 ft) — the Moody-chart default.

Absolute Roughness of Drawn Tubing and Plastic Pipe measured

\epsilon_{\mathrm{tube}} = 0.0000015\ \text{m}

mAbsolute roughness ε of drawn copper tube, glass and smooth plastic pipe, about 0.0015 mm — 30 times smoother than steel.

Absolute Roughness of Cast Iron Pipe measured

\epsilon_{\mathrm{CI}} = 0.00026\ \text{m}

mAbsolute roughness ε of new uncoated cast iron pipe, about 0.26 mm — five times rougher than steel, and far worse when tuberculated.

Hazen-Williams C for Plastic Pipe measured

C_{\mathrm{PVC}} = 150\ \text{—}

—Hazen-Williams roughness coefficient for smooth plastic pipe such as PVC, PE and CPVC, conventionally taken as 150 for design.

Hazen-Williams C for Cast Iron Pipe measured

C_{\mathrm{CI}} = 100\ \text{—}

—Hazen-Williams roughness coefficient for aged unlined cast iron pipe, typically 100 — new pipe is near 130, badly tuberculated near 60.

Manning's n for Concrete Channels measured

n_{\mathrm{concrete}} = 0.013\ \text{—}

—Manning roughness coefficient for finished concrete channels and pipe, typically 0.013 with a defensible range of 0.011 to 0.016.

Unit Weight of Loose Sand measured

\gamma_{\mathrm{sand}} = 15,500\ \text{N/m}^{3}

N/m³Typical moist bulk unit weight of loose sand, about 15.5 kN/m³ or 99 lbf/ft³; dense sand runs nearer 19.5 kN/m³.

Unit Weight of Soft Clay measured

\gamma_{\mathrm{clay}} = 16,000\ \text{N/m}^{3}

N/m³Typical saturated unit weight of soft normally consolidated clay, about 16 kN/m³ or 102 lbf/ft³, with 14–18 kN/m³ the usual range.

Specific Gravity of Soil Solids measured

G_{s} = 2.65\ \text{—}

—Specific gravity of the mineral solids in most soils, about 2.65 for quartz sands and 2.70–2.75 for clays — remarkably consistent.

Bulk Modulus of Water measured

K_{w} = 2,180,000,000\ \text{Pa}

PaIsothermal bulk modulus of liquid water at 20 °C and 1 atm, about 2.18 GPa — a 0.005 % volume change per bar of pressure.

Vapour Pressure of Water at 20 °C measured

p_{v} = 2339\ \text{Pa}

PaSaturation vapour pressure of water at 20 °C, 2339 Pa (0.339 psia) — the absolute pressure at which 20 °C water boils.

Chemistry 45

Ionic Product of Water (Kw at 25 °C) measured

K_{\mathrm{w}} = 1 \times 10^{-14}

dimensionlessThe autoionisation constant of pure water at 25 °C, [H⁺][OH⁻] = 1.0 × 10⁻¹⁴, the equilibrium behind the whole 0–14 pH scale.

pKw of Water at 25 °C measured

\mathrm{p}K_{\mathrm{w}} = 13.995

dimensionlessThe negative logarithm of water's ionic product at 25 °C, pKw = 13.995, universally rounded to 14.00 for the pH + pOH identity.

Molar Mass of Water measured

M(\mathrm{H_2O}) = 0.018015\ \text{kg/mol}

kg/molThe molar mass of water, 18.015 g/mol — one mole of H₂O is 18.015 g and occupies almost exactly 18.07 mL of liquid at 25 °C.

Molar Mass of Carbon Dioxide measured

M(\mathrm{CO_2}) = 0.044009\ \text{kg/mol}

kg/molThe molar mass of carbon dioxide, 44.009 g/mol — the conversion that turns tonnes of burnt carbon into tonnes of CO₂ emitted.

Molar Mass of Dioxygen measured

M(\mathrm{O_2}) = 0.031998\ \text{kg/mol}

kg/molThe molar mass of molecular oxygen, 31.998 g/mol — the basis of every stoichiometric air requirement in combustion calculations.

Molar Mass of Dinitrogen measured

M(\mathrm{N_2}) = 0.028014\ \text{kg/mol}

kg/molThe molar mass of molecular nitrogen, 28.014 g/mol — the dominant term in the 28.96 g/mol average molar mass of dry air.

Molar Mass of Methane measured

M(\mathrm{CH_4}) = 0.016043\ \text{kg/mol}

kg/molThe molar mass of methane, 16.043 g/mol — the working figure for natural gas, converting between cubic metres, kilograms and moles.

Molar Mass of Sodium Chloride measured

M(\mathrm{NaCl}) = 0.05844\ \text{kg/mol}

kg/molThe formula mass of sodium chloride, 58.44 g/mol — the number behind saline, brine strength and softener regeneration dosing.

Molar Mass of Calcium Carbonate measured

M(\mathrm{CaCO_3}) = 0.100086\ \text{kg/mol}

kg/molThe formula mass of calcium carbonate, 100.086 g/mol — the reference substance for reporting hardness and alkalinity as mg/L as CaCO₃.

Molar Mass of Sulfuric Acid measured

M(\mathrm{H_2SO_4}) = 0.098072\ \text{kg/mol}

kg/molThe molar mass of sulfuric acid, 98.072 g/mol — the basis for converting between percent strength, molarity and normality in acid feed.

Molar Mass of Sodium Hydroxide measured

M(\mathrm{NaOH}) = 0.039997\ \text{kg/mol}

kg/molThe formula mass of sodium hydroxide, 39.997 g/mol — the reason a 1 M caustic solution is made from almost exactly 40 g per litre.

Molar Mass of Ammonia measured

M(\mathrm{NH_3}) = 0.017031\ \text{kg/mol}

kg/molThe molar mass of ammonia, 17.031 g/mol — the conversion between mg/L as N and mg/L as NH₃ in every wastewater report.

Molar Mass of Glucose measured

M(\mathrm{C_6H_{12}O_6}) = 0.180156\ \text{kg/mol}

kg/molThe molar mass of glucose, 180.156 g/mol — the conversion between blood sugar in mg/dL and mmol/L, and the unit of cellular energy accounting.

Atomic Mass of Hydrogen measured

A_{\mathrm{r}}(\mathrm{H}) = 0.001008\ \text{kg/mol}

kg/molThe standard atomic weight of hydrogen, 1.0080 g/mol — the lightest entry on the periodic table and the anchor of the original mass scale.

Atomic Mass of Carbon measured

A_{\mathrm{r}}(\mathrm{C}) = 0.012011\ \text{kg/mol}

kg/molThe standard atomic weight of carbon, 12.011 g/mol — the element that anchored the atomic mass scale from 1961 until the 2019 redefinition.

Atomic Mass of Nitrogen measured

A_{\mathrm{r}}(\mathrm{N}) = 0.014007\ \text{kg/mol}

kg/molThe standard atomic weight of nitrogen, 14.007 g/mol — the divisor behind every fertiliser grade and every mg/L as N in a water report.

Atomic Mass of Oxygen measured

A_{\mathrm{r}}(\mathrm{O}) = 0.015999\ \text{kg/mol}

kg/molThe standard atomic weight of oxygen, 15.999 g/mol — the reference element for chemical atomic masses from Berzelius until 1961.

Atomic Mass of Sodium measured

A_{\mathrm{r}}(\mathrm{Na}) = 0.02298976928\ \text{kg/mol}

kg/molThe standard atomic weight of sodium, 22.98976928 g/mol — a mononuclidic element, so the value is known to eleven significant figures.

Atomic Mass of Chlorine measured

A_{\mathrm{r}}(\mathrm{Cl}) = 0.03545\ \text{kg/mol}

kg/molThe standard atomic weight of chlorine, 35.45 g/mol — the textbook example of a fractional atomic weight produced by isotope mixing.

Atomic Mass of Calcium measured

A_{\mathrm{r}}(\mathrm{Ca}) = 0.040078\ \text{kg/mol}

kg/molThe standard atomic weight of calcium, 40.078 g/mol — the ion that dominates water hardness and the scale it leaves behind.

Atomic Mass of Iron measured

A_{\mathrm{r}}(\mathrm{Fe}) = 0.055845\ \text{kg/mol}

kg/molThe standard atomic weight of iron, 55.845 g/mol — the element with the highest binding energy per nucleon, where fusion stops paying.

Atomic Mass of Sulfur measured

A_{\mathrm{r}}(\mathrm{S}) = 0.03206\ \text{kg/mol}

kg/molThe standard atomic weight of sulfur, 32.06 g/mol — the divisor for sulfate, sulfuric acid and every fuel sulfur specification.

Acid Dissociation Constant of Acetic Acid measured

K_{\mathrm{a}}(\mathrm{CH_3COOH}) = 0.0000175

dimensionlessThe acid dissociation constant of acetic acid at 25 °C, Kₐ = 1.75 × 10⁻⁵, the reference weak acid of every textbook and every buffer.

pKₐ of Acetic Acid measured

\mathrm{p}K_{\mathrm{a}}(\mathrm{CH_3COOH}) = 4.756

dimensionlessThe pKₐ of acetic acid at 25 °C, 4.756 — the pH at which acetic acid and acetate are present in exactly equal amounts.

First Dissociation Constant of Carbonic Acid measured

K_{\mathrm{a1}}(\mathrm{H_2CO_3}) = 4.45 \times 10^{-7}

dimensionlessThe first acid dissociation constant of carbonic acid at 25 °C, Kₐ₁ = 4.45 × 10⁻⁷, governing the CO₂-bicarbonate equilibrium in natural water.

First pKₐ of Carbonic Acid measured

\mathrm{p}K_{\mathrm{a1}}(\mathrm{H_2CO_3}) = 6.352

dimensionlessThe first pKₐ of carbonic acid at 25 °C, 6.352 — the pH at which dissolved CO₂ and bicarbonate are present in equal concentrations.

Second Dissociation Constant of Carbonic Acid measured

K_{\mathrm{a2}}(\mathrm{H_2CO_3}) = 4.69 \times 10^{-11}

dimensionlessThe second acid dissociation constant of carbonic acid at 25 °C, Kₐ₂ = 4.69 × 10⁻¹¹, the bicarbonate-to-carbonate step that drives scaling.

Second pKₐ of Carbonic Acid measured

\mathrm{p}K_{\mathrm{a2}}(\mathrm{H_2CO_3}) = 10.329

dimensionlessThe second pKₐ of carbonic acid at 25 °C, 10.329 — the pH at which bicarbonate and carbonate ions are present in equal concentrations.

First Dissociation Constant of Phosphoric Acid measured

K_{\mathrm{a1}}(\mathrm{H_3PO_4}) = 0.00711

dimensionlessThe first acid dissociation constant of phosphoric acid at 25 °C, Kₐ₁ = 7.11 × 10⁻³, a moderately strong first proton on a triprotic acid.

First pKₐ of Phosphoric Acid measured

\mathrm{p}K_{\mathrm{a1}}(\mathrm{H_3PO_4}) = 2.148

dimensionlessThe first pKₐ of phosphoric acid at 25 °C, 2.148 — the centre of the low-pH buffering region used in HPLC mobile phases.

Second Dissociation Constant of Phosphoric Acid measured

K_{\mathrm{a2}}(\mathrm{H_3PO_4}) = 6.32 \times 10^{-8}

dimensionlessThe second acid dissociation constant of phosphoric acid at 25 °C, Kₐ₂ = 6.32 × 10⁻⁸, the step that buffers cells and biological media.

Second pKₐ of Phosphoric Acid measured

\mathrm{p}K_{\mathrm{a2}}(\mathrm{H_3PO_4}) = 7.199

dimensionlessThe second pKₐ of phosphoric acid at 25 °C, 7.199 — almost exactly physiological pH, which is why phosphate buffers biology.

Third Dissociation Constant of Phosphoric Acid measured

K_{\mathrm{a3}}(\mathrm{H_3PO_4}) = 4.5 \times 10^{-13}

dimensionlessThe third acid dissociation constant of phosphoric acid at 25 °C, Kₐ₃ = 4.5 × 10⁻¹³, a proton so tightly held it needs strong alkali to remove.

Third pKₐ of Phosphoric Acid measured

\mathrm{p}K_{\mathrm{a3}}(\mathrm{H_3PO_4}) = 12.35

dimensionlessThe third pKₐ of phosphoric acid at 25 °C, 12.35 — the pH above which free orthophosphate finally becomes the dominant species.

Base Dissociation Constant of Ammonia measured

K_{\mathrm{b}}(\mathrm{NH_3}) = 0.0000177

dimensionlessThe base dissociation constant of ammonia at 25 °C, K_b = 1.77 × 10⁻⁵, making it the textbook weak base and the mirror of acetic acid.

Solubility Product of Calcium Carbonate measured

K_{\mathrm{sp}}(\mathrm{CaCO_3}) = 3.36 \times 10^{-9}

dimensionlessThe solubility product of calcite at 25 °C, K_sp = 3.36 × 10⁻⁹ — the number that decides whether a water scales or corrodes.

Solubility Product of Calcium Sulfate measured

K_{\mathrm{sp}}(\mathrm{CaSO_4}) = 0.0000493

dimensionlessThe solubility product of anhydrous calcium sulfate at 25 °C, K_sp = 4.93 × 10⁻⁵ — the gypsum scale that acid cleaning cannot remove.

Standard Hydrogen Electrode Potential exact

E^{\circ}(\mathrm{H^+}/\mathrm{H_2}) = 0\ \text{V}

VThe standard potential of the hydrogen electrode, defined as exactly 0 V at 25 °C — the zero point of the entire electrochemical series.

Standard Potential of the Zinc Half-Cell measured

E^{\circ}(\mathrm{Zn^{2+}}/\mathrm{Zn}) = -0.7618\ \text{V}

VThe standard reduction potential of Zn²⁺ + 2e⁻ → Zn at 25 °C, −0.7618 V — the anode of the Daniell cell and of every sacrificial anode.

Standard Potential of the Copper Half-Cell measured

E^{\circ}(\mathrm{Cu^{2+}}/\mathrm{Cu}) = 0.3419\ \text{V}

VThe standard reduction potential of Cu²⁺ + 2e⁻ → Cu at 25 °C, +0.3419 V — the cathode half of the Daniell cell and of copper electroplating.

Standard Potential of the Ferric–Ferrous Couple measured

E^{\circ}(\mathrm{Fe^{3+}}/\mathrm{Fe^{2+}}) = 0.771\ \text{V}

VThe standard reduction potential of Fe³⁺ + e⁻ → Fe²⁺ at 25 °C, +0.771 V — the redox couple that sets the character of natural water.

Standard Potential of the Silver Half-Cell measured

E^{\circ}(\mathrm{Ag^+}/\mathrm{Ag}) = 0.7996\ \text{V}

VThe standard reduction potential of Ag⁺ + e⁻ → Ag at 25 °C, +0.7996 V — the basis of the silver-silver chloride reference electrode.

Standard Potential of the Oxygen–Water Couple measured

E^{\circ}(\mathrm{O_2}/\mathrm{H_2O}) = 1.229\ \text{V}

VThe standard reduction potential of O₂ + 4H⁺ + 4e⁻ → 2H₂O at 25 °C, +1.229 V — the couple that drives corrosion and limits water electrolysis.

Cryoscopic Constant of Water measured

K_{\mathrm{f}} = 1.86\ \text{K}{\cdot}\text{kg/mol}

K·kg/molThe freezing-point depression constant of water, 1.86 K·kg/mol — one molal of dissolved particles lowers the freezing point by 1.86 °C.

Ebullioscopic Constant of Water measured

K_{\mathrm{b}} = 0.512\ \text{K}{\cdot}\text{kg/mol}

K·kg/molThe boiling-point elevation constant of water, 0.512 K·kg/mol — one molal of dissolved particles raises the boiling point by 0.512 °C.

Mathematical 16

Pi exact

\pi = 3.141592653589793

dimensionlessThe ratio of a circle's circumference to its diameter, 3.14159265358979 — an irrational and transcendental number defined, never measured.

Tau (2π) exact

\tau = 6.283185307179586

dimensionlessThe ratio of a circle's circumference to its radius, τ = 2π = 6.28318530717959 — one full turn, and a proposed replacement for π.

Euler's Number exact

e = 2.718281828459045

dimensionlessThe base of the natural logarithm, e = 2.71828182845905 — the unique number whose exponential function is its own derivative.

Golden Ratio exact

\varphi = 1.618033988749895

dimensionlessThe golden ratio φ = (1 + √5)/2 = 1.61803398874989 — the number that satisfies φ² = φ + 1 and the limit of Fibonacci ratios.

Euler–Mascheroni Constant exact

\gamma = 0.5772156649015329

dimensionlessThe limiting gap between the harmonic series and the natural logarithm, γ = 0.577215664901533 — still not known to be irrational.

Square Root of Two exact

\sqrt{2} = 1.4142135623730951

dimensionlessThe diagonal of a unit square, √2 = 1.41421356237310 — the first number ever proved irrational, and the ratio behind A4 paper.

Square Root of Three exact

\sqrt{3} = 1.7320508075688772

dimensionlessThe constant of Theodorus, √3 = 1.73205080756888 — the height of an equilateral triangle of side 2 and the ratio in three-phase power.

Natural Logarithm of Two exact

\ln 2 = 0.6931471805599453

dimensionlessThe natural logarithm of 2, ln 2 = 0.693147180559945 — the number that converts a decay constant into a half-life.

Natural Logarithm of Ten exact

\ln 10 = 2.302585092994046

dimensionlessThe natural logarithm of 10, ln 10 = 2.30258509299405 — the factor that converts between natural and common logarithms.

Common Logarithm of e exact

\log_{10} e = 0.4342944819032518

dimensionlessThe base-10 logarithm of e, 0.434294481903252 — the modulus of common logarithms, and the reciprocal of ln 10.

Degrees per Radian exact

\frac{180}{\pi} = 57.29577951308232\ ^{\circ}\text{/rad}

°/radThe size of one radian in degrees, 57.2957795130823° — an exact conversion factor, since the degree is defined as exactly π/180 radians.

Radians per Degree exact

\frac{\pi}{180} = 0.017453292519943295\ \text{rad}

radOne degree expressed in radians, π/180 = 0.0174532925199433 rad — the exact factor for converting degrees into radians.

Gradians per Degree exact

\frac{10}{9} = 1.1111111111111112\ \text{gon/}^{\circ}

gon/°The exact ratio of gradians to degrees, 10/9 = 1.11111111111111 — a right angle is 100 gon, so 90° maps onto 100 gon.

Catalan's Constant exact

G = 0.915965594177219

dimensionlessCatalan's constant G = 0.915965594177219 — the alternating sum of reciprocal odd squares, of unknown irrationality after 160 years.

Apéry's Constant exact

\zeta(3) = 1.2020569031595942

dimensionlessApéry's constant ζ(3) = 1.20205690315959 — the sum of reciprocal cubes, proved irrational in 1978 and still with no closed form.

Feigenbaum Delta exact

\delta = 4.66920160910299

dimensionlessThe Feigenbaum bifurcation constant δ = 4.66920160910299 — the universal ratio at which period doubling accelerates into chaos.